I experience a lot of difficulty in explaining the waveforms of an RC differentiation circuit.
Consider a battery of voltage V connected through a switch to a capacitor C and one end A of a resistor R. The other free end B of the resistor R is connected to the negative terminal of the battery.
When the switch is closed, a sharp rise in voltage (A with respect to B) is observed across the resistor.
Most of the text books I referred to state that since "the voltage across the capacitor cannot change instantaneously provided the current remains finite" one observes the sharp rise in voltage across the resistor.
Refer "Pulse, Digital and Switching Waveforms" by Jacob Millman and Herbert Taub.
When students ask me for an explanation of what is happening across the capacitor involving potentials and charges, I experience difficulties.
Can someone provide a reasonable explanation using potentials and charge movement or one at a more microscopic level ?

You can explain the charge and the voltage with
Q=CV;
dQ/dt=I;
I=CdV/dt;

With those equation, You don't need any microscopic level of explanation. With those equation You'll be able to describe the charges and the potentials across the capacitor time by time. The statement "the voltage across the capacitor cannot change instantaneously" is true because the actual sistem always has inductance. The inductance components in your system (Battery, switch, capacitor and resistor) apear in capacitor, battery, switch, resistor, and the wiring. If You include the inductance model on your circuit, You'll be able to describe the charges and the potentials across the capacitor immediatelly after the switch has been closed, there will be no discontinuities of charge and potential movement (change) from it's initial value.